1. Field of the Invention
The invention relates to a device and methods to determine rheological properties of a fluid, such as absolute viscosity. More particularly, the fluid flows through a micro-fluidic sensor, e.g., micro-fluidic device. A Micro Electro Mechanical System (MEMS) device that is a component of the micro-fluidic sensor measures a pressure drop and, optionally, a flow rate. Utilizing this data, a processor can calculate rheological properties.
2. Description of the Related Art
Fluid analysis is important in the oil field industry. Construction decisions for a new well are largely based on measurements of fluid properties, as performed down hole (directly on reservoir fluids) or up hole (on the surface). Information regarding the chemical composition, phase diagram, density and viscosity of the fluid are critical to deciding which zones of a particular well are economical to produce and to installing the right infrastructure for that zone. Other industries are also interested in monitoring various parameters of a fluid, such as viscosity, to assure, for example, proper lubrication of certain parts or to control fabrication processes.
Viscosity is the internal resistance to shear or flow exhibited by a fluid and is a measure of the adhesive/cohesive or frictional fluid property. The resistance is caused by intermolecular friction exerted when layers of fluids attempt to slide by one another. “Apparent viscosity” is frequently used for quality control purposes. The viscosity of a test fluid is compared to the viscosity of a control fluid in a common piece of equipment, for example, the torque necessary to rotate a spindle in the fluid may be compared. However, the measurements only have meaning for the specific test equipment and over a narrow range of viscosities.
“Absolute viscosity” is the tangential force per unit area required to move one horizontal plane with respect to another at unit velocity when maintained a unit distance apart by the fluid. This can be expressed mathematically by the formula:η=τw/{dot over (γ)}  (1)where η is the viscosity, τW is the shear stress applied at one of the horizontal planes and {dot over (γ)} is the shear rate of the fluid at the same plane. Absolute viscosity, while more difficult to measure, provides a more meaningful result and is independent of test equipment and method. Kinematic viscosity is a derived quantity which can be obtained by dividing the absolute viscosity of a fluid by its mass density. In some instances, sensors may directly measure the kinematic viscosity, such as the tortional plumb bob viscometer, “Viscolite” sold by Hydramotion Ltd of York, England.
Fluid flow may by “Newtonian” or “non-Newtonian.” A fluid is Newtonian if the viscous stress tensor is linearly proportional to the shear rate tensor, with viscosity being the constant of proportionality. A fluid is non-Newtonian otherwise, due to either viscosity that is not constant with shear rate, normal stress differences, or some combination of both effects.
For an incompressible Newtonian fluid flowing in the fully developed laminar regime through a tube of circular cross-section with radius R, the shear stress, τW, and apparent shear rate, {dot over (γ)}A, at the wall of the tube can be simply related to the pressure drop, Δp that develops along the tube of length, L, and respectively the volumetric flow rate, Q, by the following formulae (see reference: Rheological Techniques, 2nd edition, R. W. Whorlow):τw=RΔp/2L   (2){dot over (γ)}A=4Q/πR3  (3)The above definition for the wall shear stress is true for both Newtonian and non-Newtonian fluids flowing in a tube with circular symmetry in the fully developed laminar regime. The shear rate above is referred to as the apparent shear rate, as it is equal to the actual shear rate only in the case of a Newtonian fluid. These formulae, when used in conjunction with Equation (1) above, lead to an expression which relates the viscosity, η, to these same quantities:
                    η        =                              π            ⁢                                                  ⁢                          R              4                        ⁢            Δ            ⁢                                                  ⁢            p                                8            ⁢                                                  ⁢            QL                                              (        4        )            In the case of a non-circular cross-section of the tube, the numerical factor in the denominator is different from the value 8 shown above, and can be obtained either analytically for some geometries, by a numerical calculation for arbitrary geometries, or experimentally by calibration using known viscosity standards. This will be discussed in more detail in a subsequent section. Immediately below is a discussion of the appropriate interpretation to measure the viscosity of a non-Newtonian fluid with a capillary viscometer with circular cross-section.
For an incompressible non-Newtonian fluid flowing in a cylindrical tube, the shear rate at the wall is more difficult to derive as the shear rate given in Eq. 3 above is no longer exact, and needs to be calculated using the Rabinowitsch equation (same reference: Rheological Techniques, 2nd edition, R. W. Whorlow):
                                          γ            .                    W                =                                                            3                ⁢                                                                  ⁢                n                            +              1                                      4              ⁢                                                          ⁢              n                                ⁢                                    γ              .                        A                                              (        5        )            where n is defined as
                    n        =                                            ∂                              (                                  log                  ⁢                                                                          ⁢                  Δ                  ⁢                                                                          ⁢                  P                                )                                                    ∂                              (                                  log                  ⁢                                                                          ⁢                  Q                                )                                              =                                    ∂                              (                                  ln                  ⁢                                                                          ⁢                                      τ                    W                                                  )                                                    ∂                              (                                  ln                  ⁢                                                                          ⁢                                                            γ                      .                                        A                                                  )                                                                        (        6        )            {dot over (γ)}W is the actual shear rate at the wall and {dot over (γ)}A is the shear rate (4Q/□R3) that would be present if the fluid were Newtonian. τW is the shear stress at the wall and is given by RΔp/(2L). Since this expression involves a derivative, the actual wall stress can only be determined if several (Δp, Q) value pairs are known and more robustly if several pairs are known. The shear rate dependent viscosity can then be calculated for a non-Newtonian fluid as:
                    η        =                                            4              ⁢                                                          ⁢              n                                                      3                ⁢                                                                  ⁢                n                            +              1                                ⁢                                    π              ⁢                                                          ⁢                              R                4                            ⁢              Δ              ⁢                                                          ⁢              p                                      8              ⁢                                                          ⁢              QL                                                          (        7        )            
As mentioned earlier, a viscosity measurement of Newtonian fluid in a non-circular geometry requires a different analysis due to its lower degree of symmetry. A simple example would be that of a slit die where one of the lateral sizes is much larger than the other by a factor of 10 or larger, allowing one to simply treat the system as one-dimensional. This would apply in a capillary that was very shallow and very wide, both lateral dimensions being small compared to its length. In this case the shear stress at the wall is:
                              τ          W                =                              H            2                    ⁢                                    Δ              ⁢                                                          ⁢              P                        L                                              (        8        )            where H is the channel depth. Furthermore, the apparent shear rate for a Newtonian fluid is:
                                          γ            .                    A                =                              6            ⁢                                                  ⁢            Q                                WH            2                                              (        9        )            where W is the channel width. The viscosity is then calculated as:
                    η        =                                            WH              3                        ⁢            Δ            ⁢                                                  ⁢            p                                12            ⁢                                                  ⁢            QL                                              (        10        )            Once again, for a non-Newtonian fluid, a slightly more complicated analysis is necessary, which gives rise to the following shear rate at the wall surface:
                                          γ            .                    W                =                                                            2                ⁢                                                                  ⁢                n                            +              1                                      3              ⁢                                                          ⁢              n                                ⁢                                    γ              .                        A                                              (        11        )            The viscosity calculation for a slit geometry is appropriately modified for non-Newtonian fluids as:
                    η        =                                            3              ⁢                                                          ⁢              n                                                      2                ⁢                                                                  ⁢                n                            +              1                                ⁢                                                    WH                3                            ⁢              Δ              ⁢                                                          ⁢              p                                      12              ⁢                                                          ⁢              QL                                                          (        12        )            where once again n is defined by formula (6) above.
The result for the slit die can be generalized, in certain cases, to a rectangular die where the two lateral dimensions (H and W) are comparable. In this case the shear stress at the wall is:
                              τ          W                =                              (                                          Δ                ⁢                                                                  ⁢                PH                                            2                ⁢                                                                  ⁢                L                                      )                    ⁢                      1                                          H                /                W                            +              1                                                          (        13        )            For a Newtonian fluid, an analytical solution for the viscosity can be derived for all combinations of H and W:
                    η        =                              Δ            ⁢                                                  ⁢                          pH              3                        ⁢                          W              3                                            12            ⁢                                                  ⁢                                          QL                ⁡                                  (                                      H                    +                    W                                    )                                            2                        ⁢                                          f                *                            ⁡                              (                                  H                  W                                )                                                                        (        14        )            where the function ƒ*(x) is defined as:
                              f          *                      (            x            )                          =                  [                                                    (                                  1                  +                                      1                    x                                                  )                            2                        ⁢                          (                              1                -                                                      192                                                                  π                        5                                            ⁢                      x                                                        ⁢                                                            ∑                                                                        i                          =                          1                                                ,                        3                        ,                                                  5                          ⁢                          …                                                                    ∞                                        ⁢                                                                  tanh                        ⁡                                                  (                                                                                    π                              2                                                        ⁢                            ix                                                    )                                                                                            i                        5                                                                                                        )                        ⁢                                          WH                2                                            6                ⁢                Q                                              ]                                    (        15        )            and is given in the Table that follows (from Son, Polymer 48, p. 632, 2007).
TABLE 1H/Wf*0.001.00000.050.93650.100.88200.150.83510.200.79460.250.75970.300.72970.350.70400.400.68200.450.66340.500.64780.550.63480.600.62420.650.61550.700.60850.750.60320.800.59910.850.59610.900.59420.950.59311.000.5928
For non-Newtonian fluids in a rectangular geometry, there is no analytical solution, though numerical solutions have been tabulated for several ratios of H/W. Reference is once more made to Son, Polymer 48, p. 632, 2007, where the tabulation of many such solutions and the calculation algorithm Son provides enable the viscosity of non-Newtonian fluids to be measured in a geometry of rectangular dimensions.
Many methods to measure apparent or actual viscosity are available for laboratory or industrial use. A simple method is by measuring the time required for a volume of fluid to flow out an outlet with a well defined geometry, such as a Marsh funnel. Alternatively, a shear strain can be applied to a fluid by confining it between two rotating surfaces, such as two flat plates, two concentric cylinders, or a cone and a plate, while measuring the sheer stress developed. As another alternative, stress on a rotating object immersed in a fluid may be monitored. The ratio of stress to strain, normalized by the specific device geometry, provides a direct measurement of absolute viscosity. This is the principle operation of many commercial laboratory devices.
Further, flow through a pipe generates a pressure drop which can be monitored using differential pressure gauges. The flow itself is either imposed externally, such as by a volumetric pump, or monitored using various types of flow metering devices. If the flow is laminar and the fluid is Newtonian, then knowledge of the volumetric flow rate, pipe geometry and pressure drop across the pipe provides enough information to determine the viscosity of the flowing fluid. This principle is used in some commercial viscosity meters where mass flow rate is obtained, for example, from a Coriolis type flow meter. In this case, the resulting quantity is a kinematic viscosity and the absolute viscosity can be obtained only if the density is known independently.
Alternatively, the flow rate can be imposed and pressure sensors used to monitor pressure drop. By imposing several flow rates and measuring the corresponding pressure drops, some non-Newtonian aspects of the fluid can be observed. Such principles have also been applied within micro-fluidic devices. For example, the expression “Micro-fluidic” can refer to a sensor where fluid is forced into sub-millimeter channels. Typically, these channels have a diameter of from one to a few hundred microns. It should be noted that sometimes the same physical principles are used to determine another parameter, such as fluid flow rate while assuming a known viscosity. As a laminar character of the flow is important in this kind of measurement, methods have been devised to transform a flow that may be turbulent into several laminar substreams by using one or several bypasses. Thermal methods are commonly used to measuring flow rate, of particular prominence being hot wire anemometry methods.
Published U.S. Patent Application, Publication No. US 2005/0183496 A1, to Baek discloses a micro-fluidic rheometer formed from three etched layers. A cavity formed in the top layer is bonded over a mid-layer to form a channel. A bottom side of the mid-layer includes one-half of a pressure transducer in the form of a capacitor. A bottom piece has the other half of the transducer (other capacitor plate) and electrical connections, resulting in an absolute pressure gauge. Deflection of the bottom side of the channel changes the spacing between the two plates changing the electrical parameters of the capacitor. Several such absolute pressure gauges are placed along a fluidic channel in the shape of a slit. A pump is used to introduce a test liquid and the pump and or a valving system is used to control the flow rate.
The resonance curve shape of a resonator immersed in a fluid can be used to infer viscosity, and in some cases, the density of the fluid. The resonance device can be driven by mechanical, electromagnetic or piezoelectric methods, while the oscillation amplitude can be detected using the device as a transducer, using external gauges to measure strain, or using optical (e.g. interferometry) means to perform a direct measurement. Examples of vibrating sensor are disclosed in WO 2006/094694 A1 and WO 2007/077038 A1, both titled “A Density and Viscosity Sensor” and both by Schlumberger Technology B.V.
As another example of viscosity measurement metrology using a vibrating object, U.S. Pat. No. 7,194,902 to Goodwin, et al. discloses a down hole viscosity measuring system for oil wells. This system includes a tension wire that extends through a moving fluid. The quality factor of the resonance curve is a function of absolute viscosity.
U.S. Patent Publication No. US 2007/0061093 A1 to Angelescu, et al. discloses a method and apparatus to measure the flow rate of a fluid in a fluid channel. A flowing fluid first passes an injection element that injects a tracer into the fluid flow. This tracer is subsequently detected by first and second downstream sensors. Data from the injection element and the detection sensors are communicated to a processor that utilizes the time data to determine flow rate.
U.S. Pat. No. 7,194,902 and published Patent Application Nos. US2005/0183496 A1 and US2007/0061093 A1 are all incorporated by reference in their entireties herein.
While the above systems determine certain features of a fluid, such as viscosity or flow rate, or are effective to determine absolute viscosity under controlled parameters, there remains a need for a micro-fluidic system effective to determine absolute viscosity accurately, potentially at high ambient pressure, and potentially without prior knowledge of flow rate.